Logic Gates JEE Notes | EduRev

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JEE : Logic Gates JEE Notes | EduRev

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2 9
LOGIC GATES
INTRODUCTION :
? A logic gate is a digital circuit which is based on certain logical relationship between the input and the
output voltages of the circuit.
? The logic gates are built using the semiconductor diodes and transistors.
? Each logic gate is represented by its characteristic symbol.
? The operation of a logic gate is indicated in a table, known as truth table. This table contains all possible
combinations of inputs and the corresponding outputs.
? A logic gate is also represented by a Boolean algebraic expression. Boolean algebra is a method of
writing logical equations showing how an output depends upon the combination of inputs. Boolean algebra
was invented by George Boole.
BASIC LOGIC GATES
There are three basic logic gates. They are (1) OR gate (2) AND gate, and (3) NOT gate
? The OR gate :- The output of an OR gate attains the state 1 if one or more inputs attain the state 1.
Logic symbol of OR gate
A
B
Y=A+B
The Boolean expression of OR gate is Y = A + B, read as Y equals A 'OR' B.
Truth table of a two-input OR gate
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
? The AND gate :- The output of an AND gate attains the state 1 if and only if all the inputs are in state 1.
Logic symbol of AND gate
A
B
Y "
Output
Input
Y
The Boolean expression of AND gate is Y = A.B
It is read as Y equals A 'AND' B
Truth table of a two-input AND gate
A B Y
0 0 0
0 1 0
1 0 0
1 1 1
? The NOT gate : The output of a NOT gate attains the state 1 if and only if the input does not attain the
state 1.
Logic symbol of NOT gate
Y A
The Boolean expression  is Y = A , read as Y equals NOT A.
Truth table of  NOT gate
A Y
0 1
1 0
COMBINATION OF GATES :
The three basis gates (OR, AND and NOT) when connected in various combinations give us logic gates such as
NAND, NOR gates, which are the universal building blocks of digital circuits.
JEEMAIN.GURU
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2 9
LOGIC GATES
INTRODUCTION :
? A logic gate is a digital circuit which is based on certain logical relationship between the input and the
output voltages of the circuit.
? The logic gates are built using the semiconductor diodes and transistors.
? Each logic gate is represented by its characteristic symbol.
? The operation of a logic gate is indicated in a table, known as truth table. This table contains all possible
combinations of inputs and the corresponding outputs.
? A logic gate is also represented by a Boolean algebraic expression. Boolean algebra is a method of
writing logical equations showing how an output depends upon the combination of inputs. Boolean algebra
was invented by George Boole.
BASIC LOGIC GATES
There are three basic logic gates. They are (1) OR gate (2) AND gate, and (3) NOT gate
? The OR gate :- The output of an OR gate attains the state 1 if one or more inputs attain the state 1.
Logic symbol of OR gate
A
B
Y=A+B
The Boolean expression of OR gate is Y = A + B, read as Y equals A 'OR' B.
Truth table of a two-input OR gate
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
? The AND gate :- The output of an AND gate attains the state 1 if and only if all the inputs are in state 1.
Logic symbol of AND gate
A
B
Y "
Output
Input
Y
The Boolean expression of AND gate is Y = A.B
It is read as Y equals A 'AND' B
Truth table of a two-input AND gate
A B Y
0 0 0
0 1 0
1 0 0
1 1 1
? The NOT gate : The output of a NOT gate attains the state 1 if and only if the input does not attain the
state 1.
Logic symbol of NOT gate
Y A
The Boolean expression  is Y = A , read as Y equals NOT A.
Truth table of  NOT gate
A Y
0 1
1 0
COMBINATION OF GATES :
The three basis gates (OR, AND and NOT) when connected in various combinations give us logic gates such as
NAND, NOR gates, which are the universal building blocks of digital circuits.
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? The NAND gate :
Logic symbol of NAND gate
A
B
Y"
Output
Input Y
The Boolean expression of NAND gate isY = 
A.B
Truth table of a NAND gate
A B Y
0 0 1
0 1 1
1 0 1
1 1 0
? The NOR gate :
Logic symbol of NOR gate
A
B
Y 
The Boolean expression of NOR gate is Y A B ? ?
Truth table of a NOR gate
A B Y
0 0 1
0 1 0
1 0 0
1 1 0
UNIVERSAL GATES :
The NAND or NOR gate is the universal building block of all digital circuits. Repeated use of NAND gates (or
NOR gates) gives other gates. Therefore, any digital system can be achieved entirely from NAND or NOR
gates. We shall show how the repeated use of NAND (and NOR) gates will gives us different gates.
? The NOT gate from a NAND gate :- When all the inputs of a NAND gate are connected together, as
shown in the figure, we obtain a NOT gate
A Y"
A
?
A B= (A) Y
0 0 1
1 1 0
Truth table of a single 
input NAND gate
? The AND gate from a NAND gates :- If a NAND gate is followed by a NOT gate (i.e., a single input
NAND gate), the resulting circuit is an AND gate as shown in figure and truth table given show how an
AND gate has been obtained from NAND gates.
A
B
Y
Truth table
A B Y' Y
0 0 1 0
0 1 1 0
1 0 1 0
1 1 0 1
? The OR gate from  NAND gates :- If we invert the inputs A and B and then apply them to the NAND
gate, the resulting circuit is an OR gate.
Y
A
B
A
B
Truth table
A B A B Y
0 0 1 1 0
0 1 1 0 1
1 0 0 1 1
1 1 0 0 1
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2 9
LOGIC GATES
INTRODUCTION :
? A logic gate is a digital circuit which is based on certain logical relationship between the input and the
output voltages of the circuit.
? The logic gates are built using the semiconductor diodes and transistors.
? Each logic gate is represented by its characteristic symbol.
? The operation of a logic gate is indicated in a table, known as truth table. This table contains all possible
combinations of inputs and the corresponding outputs.
? A logic gate is also represented by a Boolean algebraic expression. Boolean algebra is a method of
writing logical equations showing how an output depends upon the combination of inputs. Boolean algebra
was invented by George Boole.
BASIC LOGIC GATES
There are three basic logic gates. They are (1) OR gate (2) AND gate, and (3) NOT gate
? The OR gate :- The output of an OR gate attains the state 1 if one or more inputs attain the state 1.
Logic symbol of OR gate
A
B
Y=A+B
The Boolean expression of OR gate is Y = A + B, read as Y equals A 'OR' B.
Truth table of a two-input OR gate
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
? The AND gate :- The output of an AND gate attains the state 1 if and only if all the inputs are in state 1.
Logic symbol of AND gate
A
B
Y "
Output
Input
Y
The Boolean expression of AND gate is Y = A.B
It is read as Y equals A 'AND' B
Truth table of a two-input AND gate
A B Y
0 0 0
0 1 0
1 0 0
1 1 1
? The NOT gate : The output of a NOT gate attains the state 1 if and only if the input does not attain the
state 1.
Logic symbol of NOT gate
Y A
The Boolean expression  is Y = A , read as Y equals NOT A.
Truth table of  NOT gate
A Y
0 1
1 0
COMBINATION OF GATES :
The three basis gates (OR, AND and NOT) when connected in various combinations give us logic gates such as
NAND, NOR gates, which are the universal building blocks of digital circuits.
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? The NAND gate :
Logic symbol of NAND gate
A
B
Y"
Output
Input Y
The Boolean expression of NAND gate isY = 
A.B
Truth table of a NAND gate
A B Y
0 0 1
0 1 1
1 0 1
1 1 0
? The NOR gate :
Logic symbol of NOR gate
A
B
Y 
The Boolean expression of NOR gate is Y A B ? ?
Truth table of a NOR gate
A B Y
0 0 1
0 1 0
1 0 0
1 1 0
UNIVERSAL GATES :
The NAND or NOR gate is the universal building block of all digital circuits. Repeated use of NAND gates (or
NOR gates) gives other gates. Therefore, any digital system can be achieved entirely from NAND or NOR
gates. We shall show how the repeated use of NAND (and NOR) gates will gives us different gates.
? The NOT gate from a NAND gate :- When all the inputs of a NAND gate are connected together, as
shown in the figure, we obtain a NOT gate
A Y"
A
?
A B= (A) Y
0 0 1
1 1 0
Truth table of a single 
input NAND gate
? The AND gate from a NAND gates :- If a NAND gate is followed by a NOT gate (i.e., a single input
NAND gate), the resulting circuit is an AND gate as shown in figure and truth table given show how an
AND gate has been obtained from NAND gates.
A
B
Y
Truth table
A B Y' Y
0 0 1 0
0 1 1 0
1 0 1 0
1 1 0 1
? The OR gate from  NAND gates :- If we invert the inputs A and B and then apply them to the NAND
gate, the resulting circuit is an OR gate.
Y
A
B
A
B
Truth table
A B A B Y
0 0 1 1 0
0 1 1 0 1
1 0 0 1 1
1 1 0 0 1
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? The NOT gate from NOR gates :- When all the inputs of a NOR gate are connected together as shown
in the figure, we obtain a NOT gate
A
Y
? The AND gate from NOR gates :- If we invert the inputs A and B and then apply them to the NOR
gate, the resulting circuit is an AND gate.
Y
A
B
A
B
? The OR gate from NOR gate :- If a NOR gate is followed by a single input NOR gate (NOT gate), the
resulting circuit is an OR gate.
A
B
Y
XOR AND XNOR GATES :
? The Exclusive - OR gate (XOR gate):- The output of a two-input XOR gate attains the state 1 if one
and only one input attains the state 1.
Logic symbol of XOR gate
A
B
Y
The Boolean expression of XOR gate is
Y A B A B ? ? . .
 or Y = A ?B
Truth table of a XOR gate
A B Y
0 0 0
0 1 1
1 0 1
1 1 0
? Exclusive - NOR gate (XNOR gate):- The output is in state 1 when its both inputs are the same that is,
both 0 or both 1.
Logic symbol of XNOR gate
A
B
Y
The Boolean expression of XNOR gate is Y A.B A.B ? ? or Y A B ? ? or A ? B
Truth table of a XNOR gate
A B Y
0 0 1
0 1 0
1 0 0
1 1 1
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2 9
LOGIC GATES
INTRODUCTION :
? A logic gate is a digital circuit which is based on certain logical relationship between the input and the
output voltages of the circuit.
? The logic gates are built using the semiconductor diodes and transistors.
? Each logic gate is represented by its characteristic symbol.
? The operation of a logic gate is indicated in a table, known as truth table. This table contains all possible
combinations of inputs and the corresponding outputs.
? A logic gate is also represented by a Boolean algebraic expression. Boolean algebra is a method of
writing logical equations showing how an output depends upon the combination of inputs. Boolean algebra
was invented by George Boole.
BASIC LOGIC GATES
There are three basic logic gates. They are (1) OR gate (2) AND gate, and (3) NOT gate
? The OR gate :- The output of an OR gate attains the state 1 if one or more inputs attain the state 1.
Logic symbol of OR gate
A
B
Y=A+B
The Boolean expression of OR gate is Y = A + B, read as Y equals A 'OR' B.
Truth table of a two-input OR gate
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
? The AND gate :- The output of an AND gate attains the state 1 if and only if all the inputs are in state 1.
Logic symbol of AND gate
A
B
Y "
Output
Input
Y
The Boolean expression of AND gate is Y = A.B
It is read as Y equals A 'AND' B
Truth table of a two-input AND gate
A B Y
0 0 0
0 1 0
1 0 0
1 1 1
? The NOT gate : The output of a NOT gate attains the state 1 if and only if the input does not attain the
state 1.
Logic symbol of NOT gate
Y A
The Boolean expression  is Y = A , read as Y equals NOT A.
Truth table of  NOT gate
A Y
0 1
1 0
COMBINATION OF GATES :
The three basis gates (OR, AND and NOT) when connected in various combinations give us logic gates such as
NAND, NOR gates, which are the universal building blocks of digital circuits.
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? The NAND gate :
Logic symbol of NAND gate
A
B
Y"
Output
Input Y
The Boolean expression of NAND gate isY = 
A.B
Truth table of a NAND gate
A B Y
0 0 1
0 1 1
1 0 1
1 1 0
? The NOR gate :
Logic symbol of NOR gate
A
B
Y 
The Boolean expression of NOR gate is Y A B ? ?
Truth table of a NOR gate
A B Y
0 0 1
0 1 0
1 0 0
1 1 0
UNIVERSAL GATES :
The NAND or NOR gate is the universal building block of all digital circuits. Repeated use of NAND gates (or
NOR gates) gives other gates. Therefore, any digital system can be achieved entirely from NAND or NOR
gates. We shall show how the repeated use of NAND (and NOR) gates will gives us different gates.
? The NOT gate from a NAND gate :- When all the inputs of a NAND gate are connected together, as
shown in the figure, we obtain a NOT gate
A Y"
A
?
A B= (A) Y
0 0 1
1 1 0
Truth table of a single 
input NAND gate
? The AND gate from a NAND gates :- If a NAND gate is followed by a NOT gate (i.e., a single input
NAND gate), the resulting circuit is an AND gate as shown in figure and truth table given show how an
AND gate has been obtained from NAND gates.
A
B
Y
Truth table
A B Y' Y
0 0 1 0
0 1 1 0
1 0 1 0
1 1 0 1
? The OR gate from  NAND gates :- If we invert the inputs A and B and then apply them to the NAND
gate, the resulting circuit is an OR gate.
Y
A
B
A
B
Truth table
A B A B Y
0 0 1 1 0
0 1 1 0 1
1 0 0 1 1
1 1 0 0 1
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? The NOT gate from NOR gates :- When all the inputs of a NOR gate are connected together as shown
in the figure, we obtain a NOT gate
A
Y
? The AND gate from NOR gates :- If we invert the inputs A and B and then apply them to the NOR
gate, the resulting circuit is an AND gate.
Y
A
B
A
B
? The OR gate from NOR gate :- If a NOR gate is followed by a single input NOR gate (NOT gate), the
resulting circuit is an OR gate.
A
B
Y
XOR AND XNOR GATES :
? The Exclusive - OR gate (XOR gate):- The output of a two-input XOR gate attains the state 1 if one
and only one input attains the state 1.
Logic symbol of XOR gate
A
B
Y
The Boolean expression of XOR gate is
Y A B A B ? ? . .
 or Y = A ?B
Truth table of a XOR gate
A B Y
0 0 0
0 1 1
1 0 1
1 1 0
? Exclusive - NOR gate (XNOR gate):- The output is in state 1 when its both inputs are the same that is,
both 0 or both 1.
Logic symbol of XNOR gate
A
B
Y
The Boolean expression of XNOR gate is Y A.B A.B ? ? or Y A B ? ? or A ? B
Truth table of a XNOR gate
A B Y
0 0 1
0 1 0
1 0 0
1 1 1
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LAWS OF BOOLEAN ALGEBRA
Basic OR, AND, and NOT operations are given below :
O R A N D NO T
A + 0 = A A. 0 = 0 A + A = 1
A + 1 = 1 A. 1 = A A . A = 0
A + A = A A . A = A A . A = A
Boolean algebra obeys commutative, associative and distributive laws as given below :
? Commutative laws :
A + B = B + A ;
A.B = B.A
? Associative laws :
A + (B + C) = (A + B) + C
A. (B . C) = (A. B) . C
? Distributive laws :
A. (B + C) = A.B + A.C
? Some other useful identities :
(i) A + AB = A (ii) A . (A + B) = A
(iii) A + ( A B) = A + B (iv) A. ( A + B) = A.B
(v) A +(B.C) = (A + B). (A + C) (vi) ( A + B).(A + C) = A .C + B.A +B.C
? De Morgan's theorem :
First theorem :
A B A.B ? ?
Second theorem :
A.B A B ? ?
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2 9
LOGIC GATES
INTRODUCTION :
? A logic gate is a digital circuit which is based on certain logical relationship between the input and the
output voltages of the circuit.
? The logic gates are built using the semiconductor diodes and transistors.
? Each logic gate is represented by its characteristic symbol.
? The operation of a logic gate is indicated in a table, known as truth table. This table contains all possible
combinations of inputs and the corresponding outputs.
? A logic gate is also represented by a Boolean algebraic expression. Boolean algebra is a method of
writing logical equations showing how an output depends upon the combination of inputs. Boolean algebra
was invented by George Boole.
BASIC LOGIC GATES
There are three basic logic gates. They are (1) OR gate (2) AND gate, and (3) NOT gate
? The OR gate :- The output of an OR gate attains the state 1 if one or more inputs attain the state 1.
Logic symbol of OR gate
A
B
Y=A+B
The Boolean expression of OR gate is Y = A + B, read as Y equals A 'OR' B.
Truth table of a two-input OR gate
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
? The AND gate :- The output of an AND gate attains the state 1 if and only if all the inputs are in state 1.
Logic symbol of AND gate
A
B
Y "
Output
Input
Y
The Boolean expression of AND gate is Y = A.B
It is read as Y equals A 'AND' B
Truth table of a two-input AND gate
A B Y
0 0 0
0 1 0
1 0 0
1 1 1
? The NOT gate : The output of a NOT gate attains the state 1 if and only if the input does not attain the
state 1.
Logic symbol of NOT gate
Y A
The Boolean expression  is Y = A , read as Y equals NOT A.
Truth table of  NOT gate
A Y
0 1
1 0
COMBINATION OF GATES :
The three basis gates (OR, AND and NOT) when connected in various combinations give us logic gates such as
NAND, NOR gates, which are the universal building blocks of digital circuits.
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? The NAND gate :
Logic symbol of NAND gate
A
B
Y"
Output
Input Y
The Boolean expression of NAND gate isY = 
A.B
Truth table of a NAND gate
A B Y
0 0 1
0 1 1
1 0 1
1 1 0
? The NOR gate :
Logic symbol of NOR gate
A
B
Y 
The Boolean expression of NOR gate is Y A B ? ?
Truth table of a NOR gate
A B Y
0 0 1
0 1 0
1 0 0
1 1 0
UNIVERSAL GATES :
The NAND or NOR gate is the universal building block of all digital circuits. Repeated use of NAND gates (or
NOR gates) gives other gates. Therefore, any digital system can be achieved entirely from NAND or NOR
gates. We shall show how the repeated use of NAND (and NOR) gates will gives us different gates.
? The NOT gate from a NAND gate :- When all the inputs of a NAND gate are connected together, as
shown in the figure, we obtain a NOT gate
A Y"
A
?
A B= (A) Y
0 0 1
1 1 0
Truth table of a single 
input NAND gate
? The AND gate from a NAND gates :- If a NAND gate is followed by a NOT gate (i.e., a single input
NAND gate), the resulting circuit is an AND gate as shown in figure and truth table given show how an
AND gate has been obtained from NAND gates.
A
B
Y
Truth table
A B Y' Y
0 0 1 0
0 1 1 0
1 0 1 0
1 1 0 1
? The OR gate from  NAND gates :- If we invert the inputs A and B and then apply them to the NAND
gate, the resulting circuit is an OR gate.
Y
A
B
A
B
Truth table
A B A B Y
0 0 1 1 0
0 1 1 0 1
1 0 0 1 1
1 1 0 0 1
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? The NOT gate from NOR gates :- When all the inputs of a NOR gate are connected together as shown
in the figure, we obtain a NOT gate
A
Y
? The AND gate from NOR gates :- If we invert the inputs A and B and then apply them to the NOR
gate, the resulting circuit is an AND gate.
Y
A
B
A
B
? The OR gate from NOR gate :- If a NOR gate is followed by a single input NOR gate (NOT gate), the
resulting circuit is an OR gate.
A
B
Y
XOR AND XNOR GATES :
? The Exclusive - OR gate (XOR gate):- The output of a two-input XOR gate attains the state 1 if one
and only one input attains the state 1.
Logic symbol of XOR gate
A
B
Y
The Boolean expression of XOR gate is
Y A B A B ? ? . .
 or Y = A ?B
Truth table of a XOR gate
A B Y
0 0 0
0 1 1
1 0 1
1 1 0
? Exclusive - NOR gate (XNOR gate):- The output is in state 1 when its both inputs are the same that is,
both 0 or both 1.
Logic symbol of XNOR gate
A
B
Y
The Boolean expression of XNOR gate is Y A.B A.B ? ? or Y A B ? ? or A ? B
Truth table of a XNOR gate
A B Y
0 0 1
0 1 0
1 0 0
1 1 1
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LAWS OF BOOLEAN ALGEBRA
Basic OR, AND, and NOT operations are given below :
O R A N D NO T
A + 0 = A A. 0 = 0 A + A = 1
A + 1 = 1 A. 1 = A A . A = 0
A + A = A A . A = A A . A = A
Boolean algebra obeys commutative, associative and distributive laws as given below :
? Commutative laws :
A + B = B + A ;
A.B = B.A
? Associative laws :
A + (B + C) = (A + B) + C
A. (B . C) = (A. B) . C
? Distributive laws :
A. (B + C) = A.B + A.C
? Some other useful identities :
(i) A + AB = A (ii) A . (A + B) = A
(iii) A + ( A B) = A + B (iv) A. ( A + B) = A.B
(v) A +(B.C) = (A + B). (A + C) (vi) ( A + B).(A + C) = A .C + B.A +B.C
? De Morgan's theorem :
First theorem :
A B A.B ? ?
Second theorem :
A.B A B ? ?
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SUMMARY OF LOGIC GATES
N am e s Sy mb ol Boolean Truth table Electrical Circuit diagram
Expression analog ue (Practical Realisation)
OR Y = A + B
AND Y = A. B
NOT Y = 
A
or
Inverter
NOR
Y A B ? ?
(OR +NOT)
NAND
Y A.B ?
(AND+NOT)
XOR
Y A B ? ?
(Exclusive    or
  OR)
Y A.B AB ? ?
XNOR Y = A ? B
(Exclusive or
NOR)
Y A.B A.B ? ?
or
Y A B ? ?
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
A B Y
0 0 0
0 1 0
1 0 0
1 1 1
A Y
0 1
1 0
A B Y
0 0 1
0 1 0
1 0 0
1 1 0
A B Y
0 0 1
0 1 1
1 0 1
1 1 0
A B Y
0 0 0
0 1 1
1 0 1
1 1 0
A B Y
0 0 1
0 1 0
1 0 0
1 1 1
A
B
Y 
A
B
Y"
Y
Y A
A
B
Y 
A
B
Y"
Y
A
B
Y
A
B
Y
A
B
A
Y
R
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